Final answer:
To find the area of a circular sector with a different radius, we use the proportionality of the area to the square of the radius. By establishing the constant of proportionality with the initial area and radius, we can calculate the area for the new radius by squaring the new radius and multiplying by the constant.
Step-by-step explanation:
The area of a circular sector is directly proportional to the square of the circle's radius. Given that the area of the sector is 2 square cm when the radius is 1.6 cm, we can use the relationship A = k · r^2, where A is the area, r is the radius, and k is a constant of proportionality. To find k, we set up the equation 2 = k · (1.6)^2.
To find the area of the sector with a radius of 2.7 cm, we apply the same proportionality constant to the new radius, leading to the following calculation: A = k · (2.7)^2. After solving for k using the initial conditions, we substitute it back into the equation to compute the new area.
Initial equation: 2 = k · (1.6)^2; solving for k yields k = 2 / (1.6)^2.
Finding new area: A = k · (2.7)^2. We use the previously found k value and substitute 2.7 for r to find A for the larger radius.